ethics in mathematics: a brief history of mathematical proofs

8/17/2019 Ethics in Mathematics: a Brief History of Mathematical Proofs

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ETHICS IN MATHEMATICS: A BRIEF HISTORY OF MATHEMATICAL PROOFS

1.0 Introduction

Our Public Affairs Mission at Missouri State University emphasizes three important

traits: cultural competence, community engagement, and ethical leadership. onsidering my

intentions to obtain a doctorate in mathematics and to teach college mathematics, ! must e"hibit

ethical leadership in my position. omposing this documented research report on ethics in

mathematics has shed much light on invaluable information that #ill undoubtedly inform my

future employment. ertainly, it #ould surprise no one to assert the value of an ethical system of

communicating mathematical results$ in fact, e"amining ethics in mathematics underscores the

necessity of viable mathematical results in sciences other than mathematics itself, such as

physics, chemistry, architecture, and more. Overall, ! claim t#o things above all else: that proofs

in mathematics have undergone consistent changes over the course of the last t#o millennia and

that proofs in mathematics are by nature ob%ective truths. &"amining the history of mathematical

proofs, ! #ill give pertinent historical information and anecdotes$ ! #ill supply relevant

definitions and terminology$ and ! #ill detail a brief introduction to mathematical logic. On the

#hole, therefore, ! intend #ith this documented research report to present my findings on the

ethical nature of mathematical proofs by e"amining the history of proof techni'ues.

2.0 Dicuion

2.1 Greece and Other Ancient Societies

2.1.1. From Babylon to Euclid’s &lements

&ven today, one of the most remar(able mathematical achievements in history is the

)abylonian discovery of more than a dozen Pythagorean triples *&ves ++. -ecall that the

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Pythagorean heorem states that the sum of the s'uares of the legs of a right triangle x and y

e'uals the s'uare of the hypotenuse z / or x0 1 y0 2 z 0. Using this, a Pythagorean triple is three

positive integers a, b, and c such that a0 1 b0 2 c0. On first glance, it is incredible that the

)abylonians #ere able to solve an e'uation in three variables through such elementary means,

but a deeper loo( reveals the most spectacular aspect of the )abylonian3s discoveries: they did

this before the Pythagorean heorem #as ever even con%ectured, let alone proven *++.

-ather, it #as not until one thousand years later that Pythagoras and his follo#ers / the

Pythagoreans / established and proved their theorem on right triangles. )efore the time of

Pythagoras, mathematics #as conducted intuitively: if the result of a claim seemed to hold in

general, the claim #as thought to be true *4rantz 56. One of the first radical thin(ers in

mathematics, Pythagoras rebu(ed this norm, believing instead that the mathematical sciences

need be founded in strong, ob%ective reasoning, and, further, asserting that many geometrical

results of the time could be deduced from a small list of postulates that #ere ta(en #ithout

'uestion to be true *56. !n this #ay, &uclid #ould dra# the necessary inspiration to generate the

mathematical #orld3s first attempt at a complete a"iomatization of all geometry.

&uclid the 7eometer emerged from the melting pot of ideas of his predecessor Pythagoras

t#o hundred years later, and he #ould go on to release #ith his Elements #hat remains one of

the most essential #or(s in mathematics. &uclid3s most notable efforts sought a connection

bet#een mathematics and logic in a rigorous manner, asserting that proofs follo# from

definitions and a"ioms that are ta(en as fact #ithout further 'uestion *58. 4rantz points out that

&uclid began #ith a fe# undefined terms that #ere commonly accepted #ithout %ustification and

used these to formulate his self9evident a"ioms, #ith #hich he introduced his logical e"position

in a very sophisticated and almost modern manner *5+. Of course, &uclid3s famed fifth postulate

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/ #hich states that there is a uni'ue line parallel to any given line / has for the last t#o thousand

years caused debates concerning the soundness and independence of his a"iom and, in this #ay,

has e"posed the #ea(nesses in the logic of the ancients.

2.1.2. Closing the Logical Gaps

&uclid3s Elements remains among the most astounding mathematical #or(s to resonate

through the halls of time, but even this master#or( of ancient 7reece has been sho#n / by our

contemporary standards / to contain several fla#s in logic. ommenting on this and other

mathematical follies, ieudonn; contests that the ma%ority of insufficient or erroneous proofs

throughout history have resulted from a lac( of focus on the underlying mechanics of proof

*ieudonn; 06+. )efore #e e"amine this claim, let us recall the structure of a deductive proof.

Proofs in mathematics are logical se'uences of con%ectures made from an initial

assumption / or several assumptions / and ending #ith a conclusion *06+. &ach step in a proof

should follo# directly from either our initial assumption or some (no#n a"iom or theorem.

Previous results have been sometimes invalidated by more recent commentators, ieudonn;

says, due to some confusion regarding the initial premise and a similar premise from #hich a

conclusion is dra#n #hen in fact it is the conclusion analogous to the similar premise that

follo#s *06+. Other instances sho# that, as ieudonn; observes, some proofs have employed

inferences and reasonable assumptions that had not yet been proven, resulting in falsity *06+.


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or a conclusion dra#n from previous steps. ?e define rigor as soundness and ob%ectivity of an

argument, guaranteed by strict implementation of and obedience to a"ioms and definitions.

4itcher argues that although some mathematical (no#ledge seems self9evident, it

remains necessary to prove these apparent truths in a rigorous #ay so as to ma(e our

understanding more taut and ob%ective *+@6. 4itcher asserts that as the study of mathematics

progresses, our notion of rigor li(e#ise progresses, so eventually, things that #e consider to be

rigorous at present #ill come to be considered unrigorous *+@6.

Overall, #e demand rigor #hen an argument cannot be fleshed out using elementary

steps from (no#n and factual premises and, similarly, #hen establishing a universal

mathematical language. ?e #ill no# consider the e"ample of e#ton and


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Big. 5. A Cery !nformal Proof

Source: Sidney Darris Ehttp:FF###.sciencecartoonsplus.comFinde".phpG

2.2 The Birth of Analysis and a Need for Rigor

2.2.1. e!ton" Leibniz" and Calculus

alculus has a very curious, inspiring, and rich history, but #e #ill simply begin in the

middle of the seventeenth century.


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e#ton and


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auchy, )olzano, and ?eierstrass #ere among the pioneers of the ne#ly9invented branch of

mathematics called analysis *oel 58+. auchy and the li(e realized the misleading nature of

geometry and gre# #ary of its grasp on several fields in mathematics, according to oel, and as

such, they sought to a"iomatize calculus using a more algebraic argument *588, 58+. Similarly,

Saul attests the crucial role of simple arithmetic / as opposed to the early geometric and intuitive

methods / in auchy and )olzano3s development of mathematical rigor *Saul ii. onvincingly,

Saul argues that ine'ualities #ere the fundamental tools by #hich the nineteenth century analysts

achieved their goal of precision, noting that until the time ine'ualities #ere studied and

developed in9depth by auchy, rigorous proofs of calculus #ere not often supplied *0+, 0J.

learly, auchy played an invaluable part in the development of mathematical proofs and

in the establishment of a ne# standard of rigor in mathematics. auchy is credited #ith

converting many of his contemporaries to forego the usual and mechanical computations and

casual intuition on #hich mathematics #as founded at the time in favor of a universal

mathematical language based on the ob%ectivity of a"ioms and definitions *&ves +>. &ves

portrays auchy3s body of #or( as prolific, as #ell as detailed and formal *+. ontemporaries

of auchy, on the other hand, did not all praise him #ith such esteem as &ves does$ in fact,

Schubring establishes a more realistic vie# of auchy, lending authenticity to the

mathematician3s body of #or( by establishing important criticisms that demonstrate #hy auchy

has become so important and highly regarded today *Schubring +80, +88.

&ven though auchy and his contemporaries had successfully established the discrepancy

bet#een geometry and algebra / developing a highly sophisticated and formal system of

a"iomatic proofs using arithmetic / there #as still much #or( to be done as history moved

to#ard the middle of the nineteenth century. Particularly, our notion of real numbers #ould begin

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to cause problems, and a formal definition of infinity and classification of certain sets of numbers

#ould be needed to continue to progress the state of mathematics and rigor.

2.3 God Created the Integers: Cantor ers!s "ronec#er

2.$.1. %erplexing %aradoxes o& 'n&inity

&ven today, one of the many indelible philosophical battles in history remains that

bet#een the 7erman mathematicians 7eorg antor and


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Other curious geometric parado"es reveal themselves a bit more easily than Neno3s.

Picture any straight line that e"tends itself indefinitely in both directions. ?e #ill stipulate that

our line does not inherently possess any breadth and, further, that this line in particular does not

inherently pass through any points. Picture also a semicircle of finite length laid do#n ne"t to

this line. ?e re'uire that our circle / li(e our line / must not inherently pass through any points.

)ut no#, if #e dra# three line segments of finite length originating from the center of the

semicircle, passing through the semicircle at #hat #e #ill designate as three points of

intersection, and ending on the line at #hat #e #ill li(e#ise designate as three points, #e could

sho# that a semicircle of finite length possesses the same number of points as a line of infinite

length *56. On his #eb page, Schechter provides a similar parado" that compares t#o line

segments of different lengths that someho# / contrarily / possess the same number of points.

Big. 0. A )i%ection )et#een a


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Under the long9standing misinterpretation of the properties of infinity, even the

characteristics of certain infinite collections of numbers remained shrouded in doubt and

misunderstanding. Predating antor by nearly 866 years, the !talian physicist and astronomer

7alileo 7alilei first observed the similarities bet#een the cardinality / or size / of the set of

natural numbers and the set of s'uares of natural numbers. ?e define the natural numbers to be

positive, #hole numbers. )rilliantly, 7alilei observed that you can pair off a natural number and

its s'uare / for instance, *5, 5, *0, +, *8, >, and so on / such that it appears as though there are

the same number of ob%ects that are s'uares of natural numbers as there are natural numbers *.

?e refer to this pairing as a bi%ection because every natural number refers to a uni'ue s'uare /

and so no t#o natural numbers refer to the same s'uare / and every s'uare is uni'uely

determined by a natural number, hence no s'uare is absent from our list. antor #ould later

prove that the set of natural numbers and the set of s'uares of natural numbers possess the same

cardinality$ that is, parado"ically, there are %ust as many s'uares of natural numbers as there are

natural numbers, even though the s'uares are more spread out.


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2.$.2. Cantor (isco)ers *et +heory


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same cardinality as the set in #hich it is contained is the e"act property that distinguishes infinite

sets from finite sets, as ede(ind theorized in 5@ *J=. Using this characterization, it is clear

that the set of natural numbers is infinite since at least one of its proper subsets / namely, the set

of s'uares of natural numbers / possesses the same size. antor demonstrated that the cardinality

of the natural numbers is the smallest infinite cardinality. &ven the set of rational numbers

possesses the same cardinality as the set of natural numbers. ?ith this astonishing brea(through,

he deemed this cardinality the first transfinite number Q6 , #hich is the Debre# numeral Kaleph

nullL *J=.

Ultimately, ho#ever, antor3s development of set theory proved itself to be parado"ical,

revealing the problematic nature of intuition and even further underscoring the misunderstanding

of the enigma of infinity. antor3s eponymous parado" characterizes many other ostensibly

contradictory results from the li(es of esare )urali9Borti and )ertrand -ussell *=5. onsider

the collection of all cardinal numbers / namely, the set of all infinite cardinalities. )ecause there

is no largest cardinal number, this collection is itself infinite, and in fact, the cardinality of this

collection is even larger than any of the cardinalities contained.

2.$.$. -ronecer’s Criticism

onsidering the breadth of abstraction and intricate nature of his arguments, antor3s

convoluted discoveries #ould come under heavy criticism from finitists the li(es of 4ronec(er.

&ven as early as 5@@, 4ronec(er attempted to bloc( several of antor3s publications in Crelle’s

/ournal / a famed 7erman mathematical periodical / due to a fundamental difference in

ideology: among many other of his be#ildering opinions, 4ronec(er believed neither in the

e"istence of infinity nor in the e"istence of irrational and transcendental numbers *O3onnor 8.

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-elations bet#een antor and 4ronec(er never bettered, and #hen 4ronec(er passed in 5>5,

the mathematicians remained at odds, never to settle their dispute. antor, ho#ever, continued to

e"tend #arm regards and #holehearted respect to his colleague until his untimely passing *+.

&thics in mathematics may not be better demonstrated else#here than in 4ronec(er and

antor3s infamous bic(ering. antor3s formalism / or a"iomatic mathematics based on abstract

manipulation of inherently meaningless symbols / represents a po#erful tool that

mathematicians use to overcome and understand the counterintuitive and e"tremely subtle nature

of certain mathematics, and it aids the discovery of important results such as set theory. On the

other hand, 4ronec(er3s intuitionism / or sub%ective mathematics that re%ects the notion and

importance of a"iomatic systems / e"emplifies a more practical aspect, restricting the

mathematician3s #or(9space to the tangible and constructible.


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thoroughly enough by auchy and his contemporaries *-ec( +. )y relating the studies of

calculus and arithmetic even more closely than his predecessors, 7erman mathematician -ichard

ede(ind too( it upon himself to rid analysis of its intuitive foundations entirely *+.

Oddly enough, ede(ind began his arithmetization of calculus by establishing his theory

of ede(ind cuts, #hich directly related the real number system #ith a geometric line *-ec( +.

?ith his eponymous Kcuts,L ede(ind observed the similarities bet#een the set of rational

numbers / or the set of fractions / and a tangible, straight line$ ho#ever, it became apparent after

some consideration that rational numbers cannot be e"pressed on a continuous, gap9free line,

since there are numbers on a continuous line that cannot be e"pressed as fractions / the s'uare

root of t#o, for e"ample *+. &"panding this idea further, ede(ind succeeded in e"pressing the

real number system in a completely original manner, finally relating rational and irrational

numbers and demonstrating that / using his system / proofs could then be obtained about the

real numbers that had not been possible before *J. &ssentially, ede(ind had discovered the

missing lin( bet#een geometry, algebra, and calculus, effectively freeing mathematics from its

intuitive upstart and advancing the systematization and rigorization of analysis.

2.0.2. #ilbert and the b3ecti)ity o& 4athematics


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before passing on to the ne"tL *&ves =8J. One of Dilbert3s various accomplishments included

his #or( to#ard the a"iomatization and systematization of all of mathematics / on #hich he

#or(ed alongside )ernays / called proof theory$ in 5>85, ho#ever, 7del demonstrated that

Dilbert3s efforts #ere in vain #hen he proved in KunimpeachableL style that Dilbert3s proof

theory could not hold in every field of mathematics *&ves =8+98J.

Oga#a nevertheless emphasizes Dilbert3s development of deductive rigor and his belief

in the freedom of concept9formation as a means to systematize mathematics and to establish the

preeminence of mathematics as universal and ob%ective truth that does not rely on any

epistemological assumptions *Oga#a 0, 56. One immediate conse'uence of Dilbert3s ubi'uity is

his a"iomatization of geometry, ta(ing nearly t#o9thousand9year9old notions of &uclid and

reinvigorating them #ith more sophisticated, modern foundations. !mportantly, this

establishment preceded some of the most groundbrea(ing results in physics, such as &instein3s

theory of relativity *&ves =8J. Overall, Dilbert3s grand efforts / more than anyone in history /

propelled mathematics to#ard the ultimate goal of rigor and consistency, and as such, he remains

one of the most rightfully celebrated mathematicians of all time.

2.) G*del+s Inco,(leteness Theore,s

2.5.1. +he 'mpossibility o& #ilbert’s %rogram

&ven more than his revolutionary and deeply impactful successes, oddly enough,

Dilbert3s ultimate failure to establish a set of a"ioms that could provide a foundation for all of

mathematics remains one of the most important shortcomings in scientific history, simply

because it resulted in the invaluable and indelible !ncompleteness heorems: t#o elegant results

that #ould forever change the face of mathematics. 4urt 7del / the then t#enty9five9year9old

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7erman logician / demonstrated in 5>85 #ith his Birst !ncompleteness heorem that there e"ist

arithmetic statements that are neither provable nor refutable in the a"iomatic system of Peano

arithmetic. ?ith his Second !ncompleteness heorem, 7del demonstrated that there are indeed

mathematical truths that cannot be harvested using the classical a"iomatic methods that had been

in development since the time of &uclid. ombining both of these conclusions, Dilbert3s program

/ #hich in essence aimed to establish the provability of every statement that could be

constructed in Peano arithmetic / #as rendered unrealizable *agel et al. =.

reative and elo'uent in his arguments, 7del challenged the ostensibly limitless

potential of the a"iomatic methods in a manner that no mathematician or logician had before.

Using conventional notation and meta9mathematics, 7del #as able to manipulate the calculus

of logic / or the formalized logical system consisting only of inherently meaningless mar(s and

strings of mar(s called formulas / in such a #ay as to derive a )erry parado" in the synta" of

meta9mathematics *0. Put simply, 7del #as able to ingeniously combine empty mathematical

symbols and #ords into a meta9mathematical statement that, #hen interpreted, inherently

contradicted itself. One e"ample of a simple )erry parado" follo#s.

6Behold the smallest positi)e integer not expressible in &e!er than thirteen !ords78

learly, the sentence contains fe#er than thirteen #ords / namely, it contains t#elve #ords / but

even more, these t#elve #ords characterize the very nature of the ob%ect that the statement

claims cannot be e"pressed$ in fact, the statement itself is the e"act definition of the integer that

it claims it cannot itself define. Dence, the statement lies in self9contradiction.

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Perhaps it is unsurprising to consider the amount of cunning and imagination involved in

the aforementioned argument$ ho#ever, all too fre'uently, mathematics is misunderstood as

blind, formal manipulation that lac(s e"pression. losely e"amined, 7del3s proof reveals the

often9overloo(ed artistic and linguistic aspects of mathematical proofs and, as such,

demonstrates the intrinsic limitations of philosophical arguments devoid of intuition.

2.5.2. +he 'nherent 9eanesses o& the :xiomatic 4ethod

Of course, 7del3s argument #as far more convoluted and comple" than demonstrated

above: as mentioned before, there is a noticeable air of narrative license and artistic liberty in the

!ncompleteness heorems. Upon publication, in fact, 7del3s results remained so enigmatic and

complicated that they #ere incomprehensible even to many mathematicians *8. ontrarily,

7del3s arguments #ere astonishingly rooted in simple arithmetic involving the prime

factorization property of the integers and simple sentential variables / or letters that stand in

pro"y for logical statements / in the calculus of logic, concatenated #ith the logical connectives

such as Knot,L Kor,L and Kif V thenL as #ell as useful punctuation mar(s.

onstituting the cru" of 7del3s proof are his eponymous 7del numbers. ?e assign the

first ten 7del numbers / 5, 0, 8, V, >, and 56 / to the elementary signs of our calculus %ust as

agel and e#man do *@6. learly, the 7del number of any sign, formula, or proof is uni'uely

determined, but to ma(e this characteristic more obvious, #e #ill offer an e"ample similar to

agel3s *=>, @0. ?e may assign a fe# of our first ten 7del numbers such that

W*∃ 2 +, W*2 2 J, W* s 2 @, W*T 2 , WT 2 >, W* x 2 55, and W* y 2 58, #here W* p refers to the

7del number of the sign, variable, or statement p. Using the property that every integer n

possesses a uni'ue prime factorization n 2 p5a

5 p0a

0V p a , 7del constructed each distinct 7del

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number in the same manner as #e construct the 7del number of the statement, Khere e"ists an

integer x such that x is the immediate successor of yL / or mathematically, K*∃ x* x 2 sy,L #here

∃ is the e"istential 'uantifier, x and y are integer variables, and sy denotes the immediate

successor of y. Observe that the se'uence of 7del numbers of the aforementioned statement is

, +, 55, >, , 55, J, @, 58, >$ therefore, #e have the corresponding 7del number

WK*∃ x* x 2 syLT 2 08+J55@>5558555@J5>@08580>>.

&ven better, it is possible to derive a sign, variable, or statement from a given integer / that is,

given a positive #hole number, #e can decide #hether it is a 7del number or not by chec(ing

its prime factorization *@J. Once again, #e rely on an e"ample similar to agel3s. Observe that

0+8666666 2 *=+*0+8*5J=0J 2 0=8JJ=.

Dence, the se'uence of 7del numbers in the associated statement is =, J, =, and so, it turns

out that the statement defined by this 7del number is, K6 2 6L *@=. )y applying these t#o

simple results to the set of all statements generated using the elementary signs #ithin the

calculus, 7del demonstrated that every statement about the calculus can be e"pressed as a

7del number #ithin the calculus and, therefore, that a )erry parado" is obtainable *@@.

areful and intentional in his reasoning, 7del e"posed the inherent #ea(nesses in the

a"iomatic method / many of #hich stem from the mechanical and distant manipulation of

symbolic logic. uriously, 4ennedy notes the cynicism that resulted from 7del3s proof, 'uoting

von eumann: Khere is no reason to re%ect intuitionism V becauseT there is no rigorous

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%ustification for classical mathematicsL *4ennedy . )ut despite the pessimism e"pressed by a

select fe#, the 'uest for truth in mathematics presses on in modern times / an age in #hich our

proof techni'ues have become highly sophisticated and sometimes torturously comple".

2.- /a,(les of 0roofs and thics in odern athe,atics

2.;.1. Computers in Contemporary 4athematics

onsidering the fact that contemporary mathematical proofs can be hundreds of pages

and may sometimes contain more than a thousand cases, computers have become an undeniable

necessity in the modern mathematician3s toolbo" *homas 0. Often, research in mathematics

can be greatly e"pedited through the use of such computational mathematical soft#are as

Mathematica, MA?+he error o& our approximation is ? str>errorA ?J.?A

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Using the language of the soft#are and a bit of ingenuity, Python becomes an e"tremely valuable

artifact in the mathematician3s arsenal.

Of course, the above e"ample disregards the efficiency of the program / in terms of its

use of random access memory *-AM, its time9comple"ity, and so on / #hich becomes an

absolutely vital consideration #hen a program must loop itself hundreds of thousands of times,

as is the case #ith the


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heorem employs another important mathematical symbol: the factorial operator. ?e define the

positive integer Ka factorialL to be the product of the positive integer a and every integer that is

smaller than a and greater than or e'ual to one / #ritten symbolically as

aZ 2 a*a / 5*a / 0V*0*5. learly, for instance, JZ 2 *J*+*8*0*5 2 506.

)roadly, ?ilson3s heorem states that a positive integer p is prime if and only if

* p / 5Z Y *95 *mod p. Using the follo#ing code / #hich is a strict implementation of ?ilson3s

heorem / in Python #ill generate primes in any desired domain.

import math

beg?%lease enter the positi)e integer greater than or eKual to 2 at !hich

you !ould lie to beginning your prime search@ ?A

begbeg


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this is not the case *)a"ter 5. Put e"plicitly, the


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nonetheless provided valuable results in graph theory, an acceptable proof of 7uthrie3s

observation eluded the mathematical community for more than 506 years / until the advent of

computers *5. One of the most controversial mathematical discoveries of the t#entieth century,

the largely computer9supplied proof of the Bour olor heorem / as the result of the #or( of the

University of !llinois3s 4enneth Appel and ?olfgang Da(en / poses an interesting and

perple"ing philosophical challenge to mathematics: if a mathematical proof cannot possibly be

chec(ed in its entirety by a human, are its results truly constituted as mathematical truth[

&ven though Appel and Da(en3s proof of the Bour olor heorem is not carried out

entirely by a machine, homas notes importantly that the parts that are computer9verified / some

5+@= graphs / cannot possibly be chec(ed by hand, and even further, the arguments that are

seemingly capable of hand9verification re'uire vastly comple" and meticulous computations, so

much so that no one has ever run the calculations *0.

uriously, homas and his team from the 7eorgia !nstitute of echnology argue / in spite

of their harsh criticisms of the Appel and Da(en formulation of the proof of the Bour olor

heorem / that the chances of a computer9generated proof resulting in error are even smaller

than the possibility of an erroneous human9constructed proof. On the other hand, ho#ever,

homas admits that it ta(es much more man9po#er to verify the correctness of a computer9

generated proof than it does to verify a proof provided by a human *J.

ertainly, the Bour olor heorem has revealed a perple"ing and controversial aspect of

mathematical proofs in the modern era. Bollo#ing homas, ! remain confident in the ability of

the computer systems that #e have built to produce reliable, testable, uniform results.

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2. hy Is Rigor I,(ortant4

Of any development in history, the tireless efforts of mathematicians as far bac( as 866

) to establish a threshold of mathematical rigor mar( easily one of the greatest / and most

essential / human con'uests ever endeavored.

Undoubtedly, #e need rigor to create a foundation of certainty and ob%ectivity in the

physical sciences. -igor in mathematics ensures that our intuitive reasoning is ob%ectively

founded. -igor enables us to say #ith certainty / barring human error / that our hypothesized,

testable results #ill hold universally / and every time / if an e"periment is conducted under the

same parameters. -igor rules out the possibility of #ild variation and chaos.

-igor is our only defense against the inevitable entropy of the Universe.

!.0 Conc"uion

Overall, based on my research, ! find it apparent that our current system of mathematical

rigor is pristine, according to current standards. onsidering every aspect of this documented

research report, ! believe firmly that our current logical foundations allo# for the most efficient

and ob%ectively true reasoning in mathematics and science.

ommenting on the Brench mathematician Pierre uhem, Ioseph Iesseph observes and

empathizes #ith uhem3s assertion that the history of mathematics has been a long, additive

process in #hich past results are built upon but not challenged *Iesseph ++>. ontrary to

uhem3s philosophy, ho#ever, Iesseph admits a fla# in uhem3s conception that mathematical

rigor has been unchanging since the time of the 7ree(s and )abylonians. -ather, Iesseph asserts

that the la#s of logic have been 'uestioned time and again by more recent commentators and that

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the arguments such critics have made have been metaphysical in nature *+J6. &ven more,

Iesseph states very clearly that he believes that no measure of mathematical rigor e"isted up to

and including the time of the mid9eighteenth century, and he further suggests that the

mathematical community give up on the idea of an immutable, unchanging standard of

mathematical rigor in favor of a more fluid foundation *+J0, +J8.

?hile ! support Iesseph3s interpretation and understanding of mathematical rigor and

confirm his observations on the history of mathematical rigor, ! must disagree #ith his

pessimistic vie# of the potential of mathematical proofs. Unrelentingly, ! remain most optimistic

in hope that the future #ill continue to unravel further the mysteries of current mathematical

conundrums and, thus, #ill gro# upon the present strength of the system.

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#or$ Cit%d

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